Mr. Lake's Math Site

Solving With Ugly Fractions

(aka Rational Functions)

The general rule I tell all of my high school students for dealing with ugly fraction equations:

Multiply everything by the bottoms (denominators) of the fractions and get rid of the fractions, then do the regular solving steps.

When we multiply everything by the bottoms of the fractions, we can cancel out the bottom of the fractions and make our problem less ugly.

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This one is even uglier.  The bottom of the fraction on the right is a quadratic, and can be factored.  If you factor it, it is going to be the same as our bottoms on the left side.  So for this one, we should factor the denominator on the right.  Then we should multiply everything by (x-7).  Then we should multiply everything by (x-7).  When we multiply, I don't actually simplify until after cancelling everything out.  The video will show this better than my typing can here.

This is kind of ugly, but we can multiply everything by 7.  That get rid of the first fraction.  

Then we can multiply everything by (x-3).  That gets rid of the second fraction.

Then we can multiply everything by 56.  That gets rid of the last fraction.

​This is pretty ugly, but not as bad.  The big thing to notice here is the difference of squares.  The people who make the SAT and the TSI and all of the College Board's tests really really like difference of squares problems.  If we factor the denominator on the right, we will get

(x+2)(x-2). 

In order to get rid of all of our fractions here, we can multiply everything by (x+2) and multiply everything by (x-2).

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